On the Berger conjecture for manifolds all of whose geodesics are closed

A conjecture of Berger states that, for any simply connected Riemannian manifold all of whose geodesics are closed, all prime geodesics have the same length. We firstly show that the energy function on the free loop space of such a manifold is a perfect Morse-Bott function with respect to a suitable cohomology. Secondly we explain when the negative bundles along the critical manifolds are orientable. These two general results then lead to a solution of Berger's conjecture when the underlying manifold is a sphere of dimension at least four.